Let R be the region enclosed by #f(x) = sinx, g(x) =1-x, and x=0#. What is the volume of the solid produced by revolving R around the x-axis?
Please see below.
Here is a picture of the region with a slice taken perpendicular to the axis of rotation.
The volume of the solid is
# = pi int_0^c ((1-x)^2 - 1/2(1-cos(2x)) dx#
# = pi[ -(1-x)^3/3-1/2x +1/2sinxcosx]_0^c#
# = pi(-(1-c)^3/3+c/2+1/2sin(c) cos(c)+1/3)#
If desired, we can rewrite using
Or we can evaluate using